Numerical modeling of extreme rogue waves generated by directional energy focusing

Three-dimensional (3D) directional wave focusing is one of the mechanisms that contributes to the generation of extreme waves, also known as rogue waves, in the ocean. To simulate and analyze this phenomenon, we generate extreme waves in a 3D numerical wave tank (NWT), by specifying the motion of a snake wavemaker. The NWT solves fully non-linear potential flow equations with a free surface, using a high-order boundary element method and a mixed Eulerian–Lagrangian time updating. Some numerical aspects of the NWT were recently improved, such as the accurate computation of higher-order derivatives on the free surface and the implementation of a fast multipole algorithm in the spatial solver. The former has allowed the accurate simulation of 3D overturning waves and the latter has led to at least a one-order of magnitude increase in the NWT computational efficiency. This made it possible to generate finely resolved 3D focused overturning waves and analyze their geometry and kinematics. In this paper, we first summarize the NWT equations and numerical methods. We then introduce a typical simulation of an overturning rogue wave, and analyze the sensitivity of its geometry and kinematics to water depth and maximum angle of directional energy focusing. We find that an overturning rogue wave can have different properties depending on whether it is in the focusing or defocusing phase at breaking onset. The maximum focusing angle and the water depth largely control this situation, and therefore the main features of the rogue wave crest, such as its 3D shape and kinematics.